Queues Applied to Telecoms

6Resource Requests and Activity

Waiting is also a kind of activity.

Cesare Pavese (1908–1950)

With this citation from the Italian author Cesare Pavese, we introduce this chapter devoted to resource requests and activity in any system. We consider continuous time and discrete time.

6.1. Infinite number of sources

6.1.1. Distribution of requests in continuous time

6.1.1.1. Poisson process of requests

In section 3.1.2, we showed that the process of appearance of rare events in a population of infinite size is a Poisson process.

A request can be considered a rare event since the appearance of a request at a given instant is low. The appearance of requests made by an infinite number of sources is therefore a Poisson process. We call them Poissonian requests.

This deduction necessitates the following a priori hypotheses:

• – Hypothesis 1: in accordance with the amnesia of the Poisson process mentioned in Proposition 1.7, the probability of appearance of x requests in the interval between time t and t +∆t depends only on the duration ∆t of this interval of time, but not the starting time t. Moreover, the appearance of a new request is entirely fortuitous, without any relationship to what has happened in the past.
• – Hypothesis 2: in accordance with the stationary principle of the Poisson process mentioned Proposition 1.6, there is no correlation between the number of requests received during different intervals of time.
• – Hypothesis 3: in accordance with the simplicity of the Poisson ...